Free math worksheets

- How many equal slices make the whole pizza? - A fraction tells how many equal parts of a whole are being described. - How many slices remain after Lucas and Sarah eat theirs? - Can any fraction be written in an equivalent simpler form?

1. The whole pizza has \(8\) equal slices. 2. Lucas eats \(2\) of the \(8\) slices, so his fraction is \(\frac{2}{8} = \frac{1}{4}\). 3. Sarah eats \(1\) of the \(8\) slices, so her fraction is \(\frac{1}{8}\). 4. The parents receive \(8 - 2 - 1 = 5\) slices, so their fraction is \(\frac{5}{8}\).

Lucas: \(\frac{1}{4}\) (or \(\frac{2}{8}\)) Sarah: \(\frac{1}{8}\) Parents: \(\frac{5}{8}\)

- Count all the sections to find the denominator. - The number of shaded sections gives the numerator for the shaded fraction. - For the unshaded fraction, determine how many sections remain.

1. Count all the sections: the strip has \(6\) equal sections. 2. Count the shaded sections: \(2\) sections are shaded. 3. The shaded fraction is \(\frac{2}{6} = \frac{1}{3}\). 4. The number of unshaded sections is \(6 - 2 = 4\). 5. The unshaded fraction is \(\frac{4}{6} = \frac{2}{3}\).

a) \(6\) b) \(2\) c) \(\frac{2}{6}\), or \(\frac{1}{3}\) d) \(\frac{4}{6}\), or \(\frac{2}{3}\)

- Count all equal parts in each figure to determine its denominator. - Count the shaded parts for the shaded numerator. - The unshaded numerator comes from the equal parts that remain. - Check that the shaded and unshaded counts together make the whole figure.

1. Figure a) has \(8\) equal parts, with \(3\) shaded. The shaded fraction is \(\frac{3}{8}\), and the unshaded fraction is \(\frac{5}{8}\). 2. Figure b) has \(4\) equal parts, with \(1\) shaded. The shaded fraction is \(\frac{1}{4}\), and the unshaded fraction is \(\frac{3}{4}\). 3. Figure c) has \(6\) equal parts, with \(4\) shaded. The shaded fraction is \(\frac{4}{6}\), and the unshaded fraction is \(\frac{2}{6}\). 4. Figure d) has \(6\) equal parts, with \(5\) shaded. The shaded fraction is \(\frac{5}{6}\), and the unshaded fraction is \(\frac{1}{6}\).

a) shaded: \(\frac{3}{8}\); unshaded: \(\frac{5}{8}\) b) shaded: \(\frac{1}{4}\); unshaded: \(\frac{3}{4}\) c) shaded: \(\frac{4}{6}\); unshaded: \(\frac{2}{6}\) d) shaded: \(\frac{5}{6}\); unshaded: \(\frac{1}{6}\)

- Count the small equal parts and the shaded parts in the model. - Think about making two equal groups from all of the small parts. - Decide how many of those two groups are completely shaded.

1. The bar has \(6\) equal parts, with \(3\) shaded. 2. Regrouping the \(6\) parts into \(2\) equal large parts puts \(3\) small parts in each half. 3. The shaded \(3\) sixths fill \(1\) of the \(2\) large parts. 4. Therefore, \(\frac{3}{6}=\frac{1}{2}\).

\(\frac{1}{2}\) of the bar is shaded.

- Count the equal parts and shaded parts in each bar. - Compare where the shaded region ends in the two same-size bars. - Write the two fractions as an equality if the shaded lengths match.

1. Bar a) has \(1\) of \(4\) equal parts shaded, so it shows \(\frac{1}{4}\). 2. Bar b) has \(2\) of \(8\) equal parts shaded, so it shows \(\frac{2}{8}\). 3. The shaded lengths are the same because splitting one fourth into \(2\) equal pieces makes two eighths. 4. Therefore, \(\frac{1}{4}=\frac{2}{8}\).

Yes. \(\frac{1}{4}=\frac{2}{8}\).

- First read the original fraction from the model. - Think about what happens to every original part when each is split in two. - Count total small parts and shaded small parts after the split.

1. The model has \(3\) equal sections, with \(1\) shaded, so the original fraction is \(\frac{1}{3}\). 2. Splitting each of the \(3\) sections into \(2\) parts makes \(3\times2=6\) small parts. 3. The one shaded third becomes \(2\) shaded small parts. 4. The new fraction is \(\frac{2}{6}\), so \(\frac{1}{3}=\frac{2}{6}\).

There would be \(6\) small parts, with \(2\) shaded. \(\frac{1}{3}=\frac{2}{6}\).

- Read the shaded fraction in model a) from its equal parts. - Compare how the same-size whole is partitioned in model b). - Think about how many smaller parts fit inside each shaded part of model a).

1. Model a) shows \(2\) of \(3\) equal parts shaded, or \(\frac{2}{3}\). 2. Model b) divides the same-size whole into \(6\) equal parts. 3. Splitting each third into \(2\) equal pieces makes \(4\) shaded sixths. 4. Therefore, \(\frac{2}{3}=\frac{4}{6}\).

Shade \(4\) parts of model b). \(\frac{2}{3}=\frac{4}{6}\).

- First read the shaded fraction from the model. - The numerator must change from the model's shaded-part count to \(6\) without changing the shaded amount. - Whatever happens to each shaded part must also happen to every part of the whole.

1. The model has \(3\) of \(4\) equal parts shaded, so it shows \(\frac{3}{4}\). 2. To change \(3\) shaded parts into \(6\) shaded parts without changing the amount, split every fourth into \(2\) equal pieces. 3. The whole then has \(8\) equal parts, and \(6\) are shaded. 4. Therefore, \(\frac{3}{4}=\frac{6}{8}\), so the denominator is \(8\).

The denominator is \(8\), giving \(\frac{6}{8}\).

- A whole contains two equal halves. - The second half needs the same number of tiles as the first. - Use the \(4\) covered tiles over the total number of tiles.

1. The other half needs the same number of tiles. 2. The whole strip needs \(4+4=8\) tiles. 3. Four of the \(8\) tiles cover one half, so the fraction is \(\frac{4}{8}\).

The whole strip takes \(8\) tiles, and the covered half is \(\frac{4}{8}\).

- For each figure, count the equal parts and the shaded parts. - Compare the actual shaded lengths because all three bars represent the same-size whole. - Look for two partitions whose shaded portions end at the same place.

1. Figure a) shows \(\frac{1}{4}\). 2. Figure b) shows \(\frac{2}{8}\), which is the same amount as \(\frac{1}{4}\). 3. Figure c) shows \(\frac{2}{6}\), which is the same amount as \(\frac{1}{3}\). 4. Figures a) and b) show the same shaded amount, while figure c) shows a different amount.

Figure c). Figures a) and b) both show \(\frac{1}{4}\) of the same-size whole, while figure c) shows \(\frac{1}{3}\).

- Name the shaded fraction in each model before changing anything. - Compare where the shaded region ends in the two same-size circles. - Think about how many sixth-size parts cover the same amount as the shaded thirds in model a).

1. Model a) has \(2\) of \(3\) equal parts shaded, so it shows \(\frac{2}{3}\). 2. Model b) has \(3\) of \(6\) equal parts shaded. 3. The shaded region in model a) reaches through \(4\) sixth-size parts of the same-size circle, so model b) needs \(1\) additional shaded part. 4. After that change, model b) shows \(\frac{4}{6}\), so \(\frac{2}{3}=\frac{4}{6}\).

One additional part of model b) must be shaded; \(\frac{2}{3}=\frac{4}{6}\).

- Count all equal cells and the shaded cells. - Then look at each vertical column as a larger equal part of the same rectangle. - Check whether both fraction names refer to exactly the same shaded region.

1. The grid has \(6\) equal cells, with \(4\) shaded, so the cell fraction is \(\frac{4}{6}\). 2. The grid has \(3\) equal columns, with \(2\) complete columns shaded, so the column fraction is \(\frac{2}{3}\). 3. Both fractions describe exactly the same shaded region, so \(\frac{4}{6}=\frac{2}{3}\).

By cells: \(\frac{4}{6}\) By columns: \(\frac{2}{3}\) \(\frac{4}{6}=\frac{2}{3}\)

- Count equal shaded parts in each eighths model. - Compare total shaded amount rather than the arrangement. - Decide how many eighths have the size of one fourth.

1. One fourth has the same total size as two eighths. 2. Both model B and model C shade exactly \(2\) of \(8\) equal parts. 3. The positions of equal shaded parts do not change their total area. 4. Both models show \(\frac{2}{8}=\frac{1}{4}\).

Both model B and model C are equivalent to \(\frac{1}{4}\). Each shows \(\frac{2}{8}\); whether the shaded eighths touch does not change the amount.

- Read the shaded fraction in each model. - Compare where each shaded region ends on the same-size bars. - For a pair that appears to match, explain how one partition can be split into the other partition without changing the shaded amount.

1. Model a) shows \(\frac{2}{3}\), model b) shows \(\frac{4}{6}\), and model c) shows \(\frac{5}{8}\). 2. The shaded region in models a) and b) ends at the same place on the same-size whole. 3. Each third in model a) can be split into \(2\) equal sixths, so \(2\) shaded thirds become \(4\) shaded sixths. 4. Therefore, \(\frac{2}{3}=\frac{4}{6}\); model c) shows a different shaded amount.

Models a) and b) are equivalent: \(\frac{2}{3}=\frac{4}{6}\).

- Read the shaded fraction in each model. - Ask what changes, and what stays the same, when a shaded part is only moved. - Compare the total shaded lengths of the same-size bars rather than the positions of individual shaded pieces.

1. Model a) shows \(\frac{2}{3}\), and model b) shows \(\frac{3}{6}\). 2. Moving a shaded sixth changes its position but not the number of shaded sixths, so model b) still shows \(\frac{3}{6}\). 3. The same-size bars show different shaded lengths: \(\frac{2}{3}\) reaches farther than \(\frac{3}{6}\). 4. Therefore, moving one shaded part cannot make the models equivalent.

No. Moving a shaded part does not change the total shaded amount, so model b) still shows \(\frac{3}{6}\), which is not equivalent to \(\frac{2}{3}\).

- Count the equal parts in the bar and the parts that are shaded. - On the number line, first determine how many equal spaces make one whole. - Name \(P\) and \(Q\) before deciding which one represents the same amount as the bar.

1. The bar has \(3\) of \(4\) equal parts shaded, so it represents \(\frac{3}{4}\). 2. The number line from \(0\) to \(1\) has \(8\) equal spaces. Point \(P\) is \(5\) spaces from \(0\), so \(P=\frac{5}{8}\). 3. Point \(Q\) is \(6\) spaces from \(0\), so \(Q=\frac{6}{8}\). 4. Six eighths cover the same amount of one whole as three fourths, so \(\frac{3}{4}=\frac{6}{8}\). Therefore, point \(Q\) matches the shaded bar.

The bar shows \(\frac{3}{4}\). Point \(P=\frac{5}{8}\) and point \(Q=\frac{6}{8}\). Point \(Q\) matches the bar because \(\frac{3}{4}=\frac{6}{8}\).

- Find the total represented by each chart before naming any slice as a fraction. - Write each color's value over the total for its own chart. - Compare corresponding slices by asking whether they cover the same share of their wholes.

1. Red is the largest slice in both charts. 2. In a), the total is \(2 + 1 + 1 = 4\). The red, blue, and green slices are \(\frac{2}{4}\), \(\frac{1}{4}\), and \(\frac{1}{4}\). 3. In b), the total is \(4 + 2 + 2 = 8\). The corresponding slices are \(\frac{4}{8}\), \(\frac{2}{8}\), and \(\frac{2}{8}\). 4. The red slices both represent one half of their chart, and each smaller slice in b) is two eighths, the same amount of its whole as one fourth in a). Therefore, the charts have the same proportions.

1) Red is largest in both charts. 2) Yes. The corresponding slices represent the same fractions of their wholes: \(\frac{2}{4}=\frac{4}{8}\) and \(\frac{1}{4}=\frac{2}{8}\).

- The same section value can represent different fractions when the chart totals are different. - Find the total represented by each chart. - Write Section A as a fraction of each chart's own total before comparing the slices.

1. The total in a) is \(2 + 1 + 1 = 4\), so Section A is \(\frac{2}{4}\) of that whole. 2. The total in b) is \(2 + 4 + 2 = 8\), so Section A is \(\frac{2}{8}\) of that whole. 3. In a), Section A covers one half of the chart. In b), it covers one fourth of the chart. 4. Therefore, the slices are not the same size even though both sections have the same numerical value.

No. Section A is \(\frac{2}{4}\) of chart a) but \(\frac{2}{8}\) of chart b), so its slice is larger in chart a).

- Read the shaded fraction from the bar first. - Group the eighth-size parts into pairs. - Check whether every shaded eighth belongs to a completely shaded pair.

1. The bar has \(5\) of \(8\) equal parts shaded, so it shows \(\frac{5}{8}\). 2. Each pair of eighths makes one fourth of the bar. 3. Five shaded eighths make \(2\) complete shaded pairs, with \(1\) shaded eighth left over. 4. Therefore, the shaded amount cannot be named as a whole number of fourths.

No. Five shaded eighths make \(2\) complete pairs with \(1\) eighth left over, so \(\frac{5}{8}\) cannot be written as an equivalent fraction with denominator \(4\).

- Read the shaded and unshaded parts of model a). - Compare the size of one part in model a) with the smaller parts in model b). - Preserve the same unshaded length, then count how many parts of model b) are left shaded.

1. Model a) has \(3\) of \(4\) equal parts shaded, so \(1\) fourth is unshaded. 2. Model b) is divided into \(8\) equal parts. The same unshaded length occupies \(2\) of those eighths. 3. Therefore, \(2\) eighths should remain unshaded and \(6\) eighths should be shaded. 4. The shaded fractions are \(\frac{3}{4}=\frac{6}{8}\).

Model b) should have \(2\) eighths unshaded; the shaded equivalence is \(\frac{3}{4}=\frac{6}{8}\).

- First read the shaded fraction in each model. - Consider how each proposed one-part change affects the shaded length. - The correct change must make the two same-size bars end at the same shaded point.

1. Model a) shows \(\frac{2}{3}\), while model b) shows \(\frac{3}{6}\). 2. Adding one shaded part to model b) makes \(\frac{4}{6}\). 3. The shaded length of \(\frac{4}{6}\) matches the shaded length of \(\frac{2}{3}\), so choice a) works. 4. Removing one shaded third from model a) makes \(\frac{1}{3}\), which does not match \(\frac{3}{6}\).

a) Add one shaded part to model b), giving \(\frac{2}{3}=\frac{4}{6}\).

- Use the longer tick marks first to name \(Q\) in fourths. - Then count all of the smaller equal spaces in one whole unit. - Count how many of those smaller spaces reach from \(0\) to \(Q\).

1. Point \(Q\) is at the third fourth-size step from \(0\), so \(Q=\frac{3}{4}\). 2. There are \(6\) small equal spaces from \(0\) to \(Q\) and \(8\) such spaces in one whole unit. 3. Each small space is \(\frac{1}{8}\) of one unit, so \(Q=\frac{6}{8}\). 4. Therefore, \(\frac{3}{4}=\frac{6}{8}\).

\(Q=\frac{3}{4}=\frac{6}{8}\), and each small space is \(\frac{1}{8}\) of one unit.

- Count the shaded fourth-size parts across bars a) and b). - Count the shaded eighth-size parts across bars c) and d). - Compare how many eighth-size parts fit in one fourth-size part of the same-size whole.

1. In Model A, bar a) has \(4\) shaded fourths and bar b) has \(1\) shaded fourth, for \(\frac{5}{4}\). 2. In Model B, bar c) has \(8\) shaded eighths and bar d) has \(2\) shaded eighths, for \(\frac{10}{8}\). 3. Each fourth has the same size as \(2\) eighths. 4. Therefore, \(5\) fourths and \(10\) eighths have the same total size, so \(\frac{5}{4}=\frac{10}{8}\).

Model A shows \(\frac{5}{4}\), and Model B shows \(\frac{10}{8}\). They are equivalent: \(\frac{5}{4}=\frac{10}{8}\).